The fire broke out at the premises on the Glen Road on the evening of Friday 26th December.
Detective Sergeant Melanie McGrory said, “Police received a report that the building was on fire shortly after 9pm.
“Northern Ireland Fire and Rescue Service personnel extinguished the fire but the building has been left totally gutted.
“We are working to establish exactly how the fire started but believe it to have been started deliberately at this stage and are appealing for anyone who noticed any suspicious activity in the area of the club to contact detectives in Woodbourne on 101.”
Information can also be passed anonymously through the independent charity Crimestoppers on 0800 555 111.